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Probability Versions of Li-Yau Type Inequalities and Applications

Feng-Yu Wang, Li-Juan Cheng

Abstract

By using stochastic analysis, two probability versions of Li-Yau type inequalities are established for diffusion semigroups on a manifold possibly with (non-convex) boundary. The inequalities are explicitly given by the Bakry-Emery curvature-dimension, as well as the lower bound of the second fundamental form if the boundary exists. As applications, a number of global and local estimates are presented, which extend or improve existing ones derived for manifolds without boundary. Compared with the maximum principle technique developed in the literature, the probabilistic argument we used is more straightforward and hence considerably simpler.

Probability Versions of Li-Yau Type Inequalities and Applications

Abstract

By using stochastic analysis, two probability versions of Li-Yau type inequalities are established for diffusion semigroups on a manifold possibly with (non-convex) boundary. The inequalities are explicitly given by the Bakry-Emery curvature-dimension, as well as the lower bound of the second fundamental form if the boundary exists. As applications, a number of global and local estimates are presented, which extend or improve existing ones derived for manifolds without boundary. Compared with the maximum principle technique developed in the literature, the probabilistic argument we used is more straightforward and hence considerably simpler.

Paper Structure

This paper contains 6 sections, 7 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Global Li-Yau type estimates
  4. $\partial M$ is convex or empty
  5. $\partial M$ is non-convex
  6. Local Li-Yau Estimates

Key Result

Theorem 2.1

Assume C for some constant $n\ge m$ and a function $K\in C(M)$, and also S for some $\sigma\in C(\partial M)$ if $\partial M$ exists. Let $t>0, x\in M,$ and $(\ell_s)_{s\in [0,t]}$ be an adapted real process such that $\ell_0=1,\ell_t=0$, $\ell_s'$ exists $\text{\rm{d}} s\times \mathbb P$-a.e. on $[ Then

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • ...and 5 more